Problem 1: Defective Widgets
BinomialA factory produces widgets where 5% are defective. If you randomly select 10 widgets, what is the probability that exactly 2 are defective?
Hint: This is a Binomial distribution problem. What are n and p?
Solution
Let \(X\) be the number of defective widgets. This follows a Binomial distribution: \(X \sim \text{Binomial}(n=10, p=0.05)\).
Using the PMF: $$ P(X=2) = \binom{10}{2} (0.05)^2 (0.95)^8 $$ $$ P(X=2) = 45 \times 0.0025 \times 0.6634 \approx 0.0746 $$
Problem 2: Lottery Game
ExpectationA game costs $2 to play. You flip a coin. If Heads, you win $5. If Tails, you win nothing. What is your expected net profit?
Hint: Net profit = Expected winnings - Cost to play
Solution
Let \(X\) be the winnings. The PMF is: $$ P(X=5) = 0.5, \quad P(X=0) = 0.5 $$
Expected winnings: $$ E[X] = 5(0.5) + 0(0.5) = 2.5 $$
Net profit = $2.50 - $2.00 = $0.50
Problem 3: Email Arrivals
PoissonEmails arrive at a rate of 3 per hour. What is the probability of receiving no emails in a given hour?
Hint: Use the Poisson distribution with λ = 3. What is k?
Solution
Let \(X \sim \text{Poisson}(\lambda=3)\) be the number of emails.
Using the PMF: $$ P(X=0) = \frac{e^{-3} 3^0}{0!} = e^{-3} \approx 0.0498 $$
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Problem:
If you perform 10 trials with success probability 0.5, what is P(X = 5)?
Answer: 0.2461